On pairs of P-Adic L-Functions for weight-two modular forms

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, L^♯_p. (f, T) and L^♭_p. (f, T), for a weight-two modular form ∑ a_nqⁿ and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_p = 0. These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič-Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.